Question

Consider two coherent sources $S1$ and $S2$ producing monochromatic waves to produce interference pattern. Let the displacement of the wave produced by $S1$ be given by $Y_1=a\cos \omega t$ and the displacement by $S2$ and $Y_2=a\cos (\omega t+\varphi )$. Find out the expression for the amplitude of the resultant displacement at a point and show that the intensity at that point will be $I=4a^2{\cos }^2\varphi /2$. Hence establish the conditions for constructive and destructive interference.
(b) What is the effect on the interference fringes in Young's double slit experiment when (i) the width of the source slit is increased; (ii) the monochromatic source is replaced by a source of white light

Answer 1

(a)

Resultant displacement at the point will be

$Y=Y_1+Y_2$

$=a\cos \left(\omega t\right)+a\cos (\omega t+\varphi )$

$=2a\cos (\omega t+\varphi /2)\cos \left(\varphi /2\right)$

Intensity is square of amplitude of displacement

$I=4a^2{\cos }^2\left(\varphi /2\right)$

For constructive interference, resultant intensity is maximum.

Hence, $\varphi /2=n\pi$ where n is an integer

$\varphi =2n\pi$

For destructive interference, resultant intensity is minimum.

Hence, $\varphi /2=n\pi \pm \pi /2$

$\varphi =2n\pi \pm \pi$

(b)

(i) When the width of the slit is increased, intensity of the light waves incident increases and hence, the intensity of the fringes formed increases. However, the fringes formed are not very sharp and increasing the slit width too much, spreads the fringes in a greater area.

(ii)

When a monochromatic light source is replaced by a white light source, fringes formed at a point consists of only a single colour (wavelength). Multiple fringes of various colours are formed. Central fringe remains white as all colours constructuvely interfere there.